How did the Greeks and Romans move between their destinations?

In the case of the Ancient Greeks and their modes of transport, they primarily relied on boats. When I say, "boats", I am speaking broadly in that the Ancient Greeks could travel on large wooden ships-(if the destination was lengthy in distance) or they may have had the Ancient equivalent of a present-day water taxi or speed boat if the destination was close in proximity. The Ancient Greeks did use horses as a mode of transport, though the horse was usually reserved for either Aristocrats, Public Officials or Military Commanders. The average Ancient Greek usually walked to his or her destination and if he or she was of moderate economic means, may have had the capacity to travel to their destination(s) "by boat"-(so to speak).

The Romans also relied heavily on boats and ships as a mode of transport-(especialy during the time of the Empire). However, because Italy is larger and shaped much differently than Greece proper, the Romans were able to travel to inland destinations with greater ease than when compared to the Greeks-(whose interior landscape was largely uncultivated and not very conducive to lengthy traveling). Traveling by horseback and by chariot was probably far more common in Ancient Rome than in Ancient Greece; here are some theoretical examples:

1. If a Roman wanted to leave Rome and travel to the smaller city of Ostia-(which is about 30 miles from Rome proper), one would typically walk to the city of Ostia and then travel down the Via Ostia. The Via Appia, was a major roadway which linked Rome to the Southeastern Italian region of Calabria. If one wanted to walk or travel by horseback or by chariot, one could do so. Keep in mind that the average Roman, in ancient times, rarely left the city of Rome-(if ever). Unless one was part of the Patrician class or one was serving in the Legions-(ranging from greater Italy to the Middle East, to Northern England), the typical, average Roman resident, lived much of his or her life in Rome proper. But, like the Greeks, the Romans also had waterway transportation; so if a Roman wanted to travel to the islands of Capri or Sicily or elsewhere in the Mediterranean, he or she could have access to shipping transport-(though would have to be of at least, moderate economic means, in order to undertake such a journey).

2. If a Greek living in Athens wanted to travel to other parts of the Hellenic mainland-(before the Roman conquest of Greece proper in 146 BC/BCE), he or she would have a very difficult time in traveling beyond the region of Attica-(where Athens is located). Again, unless one was an Aristocrat, a Public Official or a Military Commander, the typical Greek resident of Athens, either remained in Athens for much of his or her life-(i.e. Socrates) or they would have access to boats and ships which could transport them to some of the cities along the mainland, the Aegean islands, Crete, the coastal cities in Asia Minor-(present-day Turkey),as well as to the Magna Grecia coastal cities in Southern Italy and Sicily. But, as in the case of the (aforementioned) Romans, the Ancient Greek resident of Athens would have to be of at least moderate economic means in order to travel beyond....The Acropolis Hill.

Compared with Contemporary society-(especially, in the United States), if you were a resident of either Ancient Rome or Athens, you pretty much spent the majority of your life in Rome or Athens proper. Waterway transport was perhaps, the one mode of transport that allowed Romans and Ancient Athenians to venture beyond their city proper or their larger city-state.

First off the Greco-Roman world extends over a long time period. Depending on how you measure it that could span a millennium. So the answer to your question changes depending on when you are travelling, (and the season of the year; some seasons you travel by sea, and some you must travel by land) and where you are travelling.

Beyond that, the answer depends on your social class. If you are a plebian, you're going to walk. If you are a slave, and IF you are permitted to travel, you're going to walk. If you are of the Senatorial class, you may ride, or you may be carried. The equestrian class were originally those who were wealthy enough to own/support a horse and armor, and notionally if you were below that threshold, you were not going to ride.

While I cannot prove it, I would be very skeptical of any claim about public transport. Public services are (in general) an artifact of the modern welfare state. Rome provided some public services (the Urban Praetor was responsible for the sewers), but generally those services were provided grudgingly and poorly. There is no fundamental public interest in speeding travel.

I suspect that Romans walked, with very few exceptions.

The following paper models potential transport, and includes speed estimates. Travel in the Greco-Roman World

Most of the travelers along Roman highways were caravans, camels, horses, and donkeys.
Bible History

Travel & Transportion in Ancient Greece is a bit older than you asked but does discuss carts and chariots.